We identify a free-abelian group from integral polytopes and its properties.
problem Understanding the structure of integral polytope groups.
method Explicitly constructing a basis and identifying an involution.
result We compute the Grothendieck group of integral polytopes and its properties.
Every symmetric polytope is a norm in the Grothendieck group of R^n polytopes.
problem Understanding norms in the Grothendieck group of polytopes.
method Grothendieck construction applied to Minkowski sum of integral polytopes.
result Symmetric polytopes are norms in the Grothendieck group for all dimensions.
Researchers prove finiteness of integral representations on specific polytopes.
problem Proving finiteness of integral representations on 2-perfect truncation polytopes.
method Analyzing the geometric component of the deformation space of properly convex real projective structures on Coxeter orbifolds.
result Contains only finitely many integral representations.
Polytopes connect Lie theory to physics, integrating integrable systems.
problem Understanding connections between Lie theory and field theories.
method Using Coxeter Plane and integrable systems, a systematic mathematical treatment.
result Supports physical proposals linking polytopes to field theories.
Analyzes vector fields in polytope decompositions, proving curve finiteness.
problem Analyzing vector fields in polytope decompositions.
method Proves integral curves are chopped into finitely many pieces by polytope decompositions.
result Finiteness of edge flips in discrete Yamabe flow.
Proves necessity of at least log2(n) layers to compute maximum of n numbers.
problem Computing the maximum of n numbers with ReLU neural networks.
method Uses lattice polytopes and duality with Newton polytopes to prove depth lower bounds.
result Proves that log2(n) hidden layers are necessary and sufficient.
Paper shows non-integrality of dike building model and provides conditions for integrality.
problem Determining the integrality of a dike building model for flood protection.
method Analyzes experimental data and mathematical proofs to establish conditions for integrality.
result Established non-integrality of the polytope and conditions for linear programming relaxation to be integral.
A metric defined on body moduli space.
problem Defining a metric on the moduli space of bodies.
method Quotient space by integral affine transformations.
result Identifies moduli space of Delzant polytopes with symplectic toric manifolds.
Study of finite energy quasiplurisubharmonic functions on toric Kähler manifolds.
problem Characterizing and understanding finite energy quasiplurisubharmonic functions on toric Kähler manifolds.
method Characterization through convex functions and integrability properties of Legendre transforms.
result Log-Lipschitz convex functions on Delzant polytopes correspond to toric quasiplurisubharmonic functions with exponential integrability.
New mathematical invariants derived from polytopes of matrices over rings.
problem Understanding Bieri-Neumann-Strebel invariants via algebraic structures.
method Investigating Newton polytopes of determinants of matrices over rings of twisted Laurent polynomials.
result Established a connection between Bieri-Neumann-Strebel invariants and Newton polytopes.
Defines universal L2-torsion for 3-manifolds, linking to polytopes.
problem Calculating L2-torsion for 3-manifolds. method Defining universal L2-torsion in terms of universal covering chain complex, studying its properties. result Identifies universal L2-torsion with polytopes and various invariants. New method improves sampling efficiency for complex distributions.
problem Sampling from distributions with high condition numbers and constraints.
method Riemannian Hamiltonian Monte Carlo with numerical integrators.
result Convergence rate is independent of condition number and polytope geometry.
Quantizes b-symplectic toric manifolds using T-modules.
problem Quantization of b-symplectic toric manifolds. method Bohr-Sommerfeld quantization via T-modules. result Dimension of quantization coincides with signed count of integral points in moment polytope.
We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as $H_5(M_{23},\ZZ)=\ZZ_7$ and $H_3(M_{24},\ZZ)=\ZZ_{12}$. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce…
Proposes a method for Gaussian Process regression on binned data.
problem Regression on binned data often leads to inaccurate predictions.
method Gaussian Process regression tailored for binned data.
result Effective probabilistic predictions of latent functions.
New flag area measures constructed via integration over differential forms.
problem Constructing new flag area measures in high-dimensional spaces.
method Using local parallel sets and integration over the normal cycle of differential forms.
result Flag area measures span the space of all smooth SO(n)-covariant flag area measures.
In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersu…
New hyperbolic manifolds discovered that fiber algebraically up to dimension 8.
problem Finding hyperbolic manifolds that fiber algebraically in all dimensions 5 to 8.
method Assigning colors and states to right-angled hyperbolic polytopes and applying arguments from Jankiewicz et al.
result First examples of hyperbolic manifolds with finitely presented but not of finite type fundamental groups.
The study broadens the concept of cyclic polytopes to Veronese polytopes.
problem Extending the framework of cyclic polytopes to a broader class of polytopes.
method Described facial structure and combinatorial characterisation of facets via σ-parity alternating sequences.
result Established a bijective correspondence between combinatorial types of Veronese polytopes and partitions of finite sets.
The Thurston norm is derived from polytopes and applied to group cohomology.
problem Understanding the structure of finitely generated torsion-free groups.
method Using the Strong Atiyah Conjecture and L2-Betti numbers, the Thurston norm is defined and related to polytopes. result The Thurston norm is a seminorm on the first cohomology group of a group with real coefficients.
The paper studies deformation spaces of Coxeter truncation polytopes.
problem Understanding the geometric properties and deformations of Coxeter truncation polytopes.
method Analyzing Coxeter truncation polytopes and their deformation spaces.
result Description of deformation spaces for Coxeter truncation polytopes of dimension d⩾4. Neural networks approximate unit spheres as polytopes.
problem Approximating unit spheres with neural networks.
method Using ReLU activation in neural networks to generate polytopes.
result Neural networks can approximate unit spheres as polytopes.
The study classifies all compact hyperbolic polytopes with eight facets.
problem Classifying compact hyperbolic Coxeter four-polytopes with specific numbers of facets.
method Complete classification through mathematical analysis.
result The complete classification of compact hyperbolic Coxeter four-polytopes with eight facets.
Two operations transform simple polytopes with same manifold structure.
problem Transforming simple polytopes while maintaining a specific manifold structure.
method Biflip and puzzle-move operations on simple polytopes.
result Produced polytopes have diffeomorphic moment-angle manifolds.
Generalizes moment-angle manifolds to arbitrary nice manifolds with corners.
problem Computing cohomology groups and rings for moment-angle manifolds.
method Stable decomposition, rim-cubicalization, partial diagonal maps, polyhedral product.
result Derived formulas for integral cohomology groups and rings of moment-angle manifolds.
Proves stability in Weyl polytopes using optimal transport.
problem Stability of Weyl polytopes under optimal transport.
method Optimal transport stability for reflexive Weyl polytopes.
result Weak metric SYZ conjecture holds for Delzant reflexive Weyl polytopes.
Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…
Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.
problem Characterizing faces of quasi-arithmetic Coxeter polytopes.
method Proof of quasi-arithmetic property of faces and sufficient condition for arithmetic faces.
result Lower-dimensional faces of quasi-arithmetic Coxeter polytopes are quasi-arithmetic.
New infinite series of hyperbolic polytopes with special growth rates found.
problem Finding new infinite series of non-compact hyperbolic polytopes.
method Constructing infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes.
result Growth rates of the constructed polytopes are Perron numbers.
This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…
Smoothly approximating convex polytopes proves dihedral rigidity.
problem Proving dihedral rigidity for convex polytopes.
method Smooth approximation approach following Brendle.
result Proved Gromov's dihedral rigidity conjecture for convex polytopes.
Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.
problem Proving smoothness of deformation space for Coxeter polytopes.
method Analyzing natural map into realization space.
result Deformation space of Coxeter 3-polytopes is smooth.
The study classifies all compact 5D polytopes with 9 facets.
problem Classifying compact hyperbolic Coxeter polytopes.
method Complete classification through mathematical analysis.
result A complete list of compact hyperbolic Coxeter 5D polytopes with 9 facets.
Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v…
The study classifies 331 specific 4D polytopes with 7 facets.
problem Classifying finite-volume hyperbolic Coxeter 4D polytopes.
method Complete classification through exhaustive search.
result 331 unique polytopes with 7 facets identified.
Contact manifolds' momentum polytopes are convex.
problem Understanding the structure of contact manifolds.
method Using isomorphism to toric varieties.
result Momentum polytopes of contact manifolds are convex.
Polytopic Matrix Factorization models data as latent vectors from a polytope, maximizing determinant for identifiability.
problem Data decomposition with semi-structured latent vectors and polytope constraints.
method Model input data as latent vectors from a polytope, using determinant maximization for identifiability.
result Identifiability condition for polytopes with specific symmetry restrictions.
The article studies factorization structures in geometry and their applications to cones and polytopes.
problem Understanding and characterizing factorization structures in geometry.
method Comprehensive study of factorization structures, including structure theory, construction of compatible polytopes and cones, and derivation of generalised Gale's evenness condition.
result Established generalised Vandermonde identities and found examples of Delzant and rational Delzant compatible polytopes.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
New cohomological rigidity results for manifolds defined by right-angled polytopes.
problem Establishing cohomological rigidity for manifolds defined by specific polytopes.
method Using techniques from toric topology, the authors prove cohomological rigidity for families of manifolds associated with polytopes from a specific class.
result Cohomology ring isomorphisms imply diffeomorphisms for manifolds in the families, and vice versa.
Examines nonrational polytopes and fans in toric geometry.
problem Understanding nonrational convex polytopes and fans in toric geometry.
method Discussion and interrelation of recent developments.
result Exploration of nonrational polytopes and fans in toric geometry.
New methods classify hyperbolic polytopes with up to 40 facets.
problem Classifying compact hyperbolic Coxeter polytopes with specific facet counts.
method New combinatorial method via point set order types.
result Proves existence of a compact hyperbolic Coxeter 29-polytope with at least 40 facets.
Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial for…
The paper explains how to parameterize facets of moment polytopes in real symplectic geometry.
problem Understanding the moment polytopes in real symplectic geometry.
method Parameterizing equations of facets of Delta(Z) in terms of real Ressayre's pairs of Z.
result Parameterization of facets of moment polytopes explained.
Unified cosmological and Einstein polytope theories.
problem Unified understanding of cosmological and Einstein polytope theories.
method Unified combinatorial perspective of cosmological and Einstein polytope theories.
result Unified construction of cosmological and Einstein polytope theories.
The study of symmetries in manifolds derived from colored polytopes.
problem Existence and types of symmetries in rational homology 3-spheres.
method Analysis of hyperbolic manifolds and right-angled polytopes.
result Described how to create colorings with specific symmetries.
Diffeomorphisms of convex polytopes form a Lie group.
problem Understanding transformations of convex polytopes.
method Forming a Lie group from diffeomorphisms of convex polytopes.
result The group of diffeomorphisms of a convex polytope is a regular Lie group.
Efficiently projects points onto polytopes, especially useful in web-scale applications.
problem Efficiently projecting points onto polytopes in large-scale applications.
method Developed a vertex-oriented incremental algorithm for polytope projection, tailored for simplex and unit-box cut polytopes.
result Majority of projections lie on vertices of polytopes, leading to significant performance improvements.